Constrained Coverage Optimisation for Mobile Cellular ... - CiteSeerX

Constrained Coverage Optimisation for Mobile Cellular Networks Lin Du, John Bigham Electronic Engineering Department Queen Mary,University of London London E1 4NS, United Kingdom {lin.du, john.bigham}@elec.qmul.ac.uk

Abstract. This paper explores the use of evolutionary algorithms to optimise cellular coverage so as to balance the traffic load over the whole mobile cellular network. A transformation of the problem space is used to remove the principal power constraint. A problem with the intuitive transformation is shown and a revised transformation with much better performance is presented. This highlights a problem with transformationbased methods in evolutionary algorithms. While the aim of transformation is to speed convergence, a bad transformation can be counterproductive. A criterion that is necessary for successful transformations is explained. Using penalty functions to manage the constraints was investigated but gave poor results. The techniques described can be used as constraint-handling method for a wide range of constrained optimisations.

1

Introduction

Mobile cellular networks are by far the most common of all public wireless communication systems. One of the basic principles is to re-use radio resources after a certain distance. The whole area is divided up into a number of small areas called cells, with one base station giving radio coverage for each cell by its associated antenna. This paper considers a novel geographic traffic load balancing scheme achieved by changing cell size and shapes, to provide dynamic mobile cellular coverage control according to different traffic conditions. Study of dynamic cell-size control has shown that the system performance can be improved for non-uniformly distributed users [1]. However changing both cell size and shapes has not been studied so far. The formation of cells is based upon call traffic needs. Capacity in a heavily loaded cell can be increased by contracting the antenna pattern around the source of peak traffic and expanding adjacent antenna patterns to fill in the coverage loss as illustrated in Fig. 1. Realising such a system requires the capability of approximately locating and tracking mobiles in order to adapt the system parameters to meet the traffic requirements. The existing generation of cellular networks has a limited capability for mobile position location, but the next generation of cellular networks is expected to have much better capabilities. The position location capabilities of the

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Fig. 1. Cellular coverage control according to geographic traffic distribution.

cellular network can be used to find the desired coverage patterns to serve a current traffic demand pattern. The best-fit antenna pattern from the available antenna resource is then synthesised using a pattern synthesis method, which takes into account the physical constraints of the antenna. The first step towards intelligent cellular coverage control is to know which combination of antenna patterns is suitable for a traffic distribution. Several optimisation methods can be used. In this work, we explore the use of genetic algorithms, one of the evolutionary algorithms, because of their robustness and efficiency in searching the optimum solution for complicated problems [2], [3], [4]. This is explained in sect. 2. Section 3 describes the transformations of the coordinate space that remove the central power constraint. While several transformations may be possible and mathematically correct, for rapid convergence care has to be taken to ensure that the transformation does not distort the space in an unsatisfactory way. A criterion that is necessary for successful transformations is also explained. Computer based simulations are performed to evaluate the system capacity. The simulations are described and results are presented in sect. 4. As a comparison, using penalty function to manage the constraints is also investigated but gave poor results. Beyond the scope of our application, the techniques described have further applications on constraints handling in optimisations. Attempts of applying this constraint-handling method into other optimisation problems are also investigated, and some of them are presented in sect. 5.

Constrained Coverage Optimisation for Mobile Cellular Networks

2

3

Using Genetic Algorithms

Genetic Algorithms are search algorithms based on the mechanics of natural selection and natural genetics. They combine survival of the fittest among string structures. In every generation, a new set of structures is created using parts of the fittest of the previous generation with occasional random variation (mutation). Whilst randomised, genetic algorithms are not simple random walks. They have been shown to efficiently exploit historical information to find new search points with expected improved performance [2]. Since genetic algorithms have already been widely used in many optimisation area, we only focus on how to apply it in our antenna patterns optimisation problem and how to handle the central power constraint. First, we must design an efficient coding scheme to represent each possible combination of antenna patterns as chromosomes. We use a gain vector, − → G = [g1 , g2 , g3 , . . . , gN ], in which each gain value is coded as a gene, symbolising antenna gains along N directions. This determines the approximate shape of one antenna pattern. The number of gains N , and the number of bits for each gain, can be determined by the performance and precision requirements. Therefore, the chromosome for a region is formed by combining each set of genes in the region. If we have M base stations, the chromosome will be of the form − → − → − → [ G 1 , G 2 , . . . , GM ]. As many researchers (e.g. [5], [6], [7], [8]) have reported, representing the optimisation parameters as numbers, rather than bit-strings, can speed up the convergence for most real-value optimisation problems. We use a real-coded genetic algorithm, with BLX-α [9] crossover, distortion crossover (intra-crossover within a subset of chromosome, i.e. crossover between two gain values at different directions in one pattern such that it can distort the shapes of individual patterns), and creep mutation operators. The elitism selection is used to prevent losing the best solutions. Since the transformation, which is explained next, handles the central constraint, it is not necessary to use more complicated operators for our optimisation. A cellular network simulator is used to calculate the fitness value for each chromosome.

3

Constraint Handling Method

The genetic algorithms are naturally an unconstrained optimisation techniques. When a new chromosome is created by crossover or mutation in the searching process, it may not be feasible. Many constraints handling approaches have already been proposed [10], [11] and recent survey papers classify them into five categories, namely: use of penalty functions, special representation and operators, separation of objectives and constraints, hybrid methods and other approaches [12]. We investigate a method based on a transformation between search space and feasible space, which ensures that all the products of a crossover or mutation always will be feasible. This falls into the second category mentioned above. It is simpler, and usually better, than the methods that only map unfeasible chromosomes to feasible ones for several reasons. i) A very simple search

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space can be used, ii) all the feasible or unfeasible chromosomes can be treated without any difference, iii) this transformation can be constructed before starting optimisation, and iv) the transformation can be done as many times as needed, if there are more than one constraint. In their paper [13] the authors propose a general mapping method, which is able to map all the points in searching space into feasible space. It is a numerical method, and involves a lot of computation in searching the boundary points. To avoid this we construct a specific mapping function by analytical means, which is explained next. The feasible space here is the space that includes all the legitimate subsets (one pattern at N directions) for a chromosome (M patterns for the whole − → network), F = { G = (g1 , . . . , gN ) : gi ∈ R}. A transformation is created such − → that the search space of the form, S = { X = (x1 , . . . , xN ) : 0 ≤ xi ≤ 1, xi ∈ R}, can be used. Each time that we need to calculate a fitness value for any − → chromosome (which is now encoded in terms of the X ), we map the chromosome into the feasible space, calculate the fitness value for new chromosome, and then assign the value to the original chromosome. In this way, we can perform genetic algorithms without constraints. According to physics theory, the radio frequency (RF) transmitting power at a base station can be expressed as, Ptrans = δ ·

N 1 X 02 g , N i=1 i

(1)

0

where N is the number of gains, gi is the i − th gain value along N directions, and δ is a constant. Then the main constraint of RF power available at the base station is expressed as, N 1 X 02 g ≤Pmax , (2) Pmin ≤ δ · N i=1 i

where Pmin is the minimum, and Pmax is the maximum value of RF power for a base station. They are determined by both physical limit and traffic density nearby. If we choose the same N for all the base stations, (2) can be simplified as, Pmin ≤ q

N δ

N X i=1

gi2 ≤ Pmax ,

0

· gi . Therefore, the feasible space F can be expressed as, ) ( N X − → F = G = (g1 , . . . , gN ) : Pmin ≤ gi2 ≤ Pmax , gi ∈ R .

where gi =

(3)

(4)

i=1

Since an N-dimensional cube is used as the search space S, n o − → S = X = (x1 , . . . , xN ) : 0 ≤ xi ≤ 1, xi ∈ R ,

(5)

Constrained Coverage Optimisation for Mobile Cellular Networks

5

we then need to define a mapping function f : S −→ F to map points from S into F.

x2

g2

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Pmax

f2

S2 0

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(a) g3

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Pmax g2

(b) Fig. 2. (a) Mapping from search space S2 to feasible space F2 , (b) Mapping from search space S3 to feasible space F3 .

When N = 2 and N = 3, search space S2 , S3 and feasible space F2 , F3 are shown in Fig. 2. Inspired by the transformation from polar coordinates and spherical coordinates to Cartesian coordinates, we can decompose the feasible space F as,  g1 = r · cosθ1     g2 = r · sinθ1 · cosθ2    g3 = r · sinθ1 · sinθ2 · cosθ3 . (6) ...     gN −1 = r · sinθ1 · sinθ2 · . . . · cosθN −1    gN = r · sinθ1 · sinθ2 · . . . · sinθN −1 P 2 Then we get gi ≡ r2 .

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If we let 

r2 = x1 · (Pmax − Pmin ) + Pmin . θi = xi+1 , i = 2, 3, . . . , N − 1

(7)

Then we get the mapping function f as,

f=

 π i=1   gi = r · cos( 2 · x2 ),  i  Q  π π   sin( 2 · xj ), 2 ≤ i ≤ N − 1  gi = r · cos( 2 · xi+1 ) j=2

N Q   gi = r · sin( π2 · xj ),    j=2  p   r = x1 · (Pmax − Pmin ) + Pmin

.

(8)

i=N

→ → → → We can prove that, ∀− x ∈ S, there exists − g = f (− x ), which obeys − g ∈ F, and vice versa. However, whilst this method is mathematically correct it has a serious problem: it distorts the space in an unsatisfactory way, which will be explained next. 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8

2

Fig. 3. The marginal PDF of gi uniformly distributed in feasible space, where N = 12, Pmax = 5.0, and Pmin = 0.01

To evaluate how much distortion the transformation causes, we use a PDF (Probability Density Function) based method, which checks the probability of output with uniformly distributed, independent input. Since the possible locations of the optimum solution are unknown in our case, the ideal transformation should generate output points uniformly distributed in feasible space. Because of the symmetry of our feasible space, the marginal PDFs of gi are identical with these of gj , where i 6= j and gi is independent of gj . One of them is shown in Fig. 3, and will be used as the criterion for evaluating transformation distortion. Whilst this method is not very fine, it works well in our cases, since a little distortion does not affect the performance of a randomised optimisation like GA.

Constrained Coverage Optimisation for Mobile Cellular Networks 6

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5 15 4 3

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Fig. 4. The marginal PDF of g1 , g4 , g9 , and g12 , using (8), where N = 12, Pmax = 5.0, and Pmin = 0.01

The marginal PDFs of g1 , g4 , g9 , and g12 , calculated by for (8) are shown in Fig. 4. It shows that the PDFs for different gi are different. g1 always has the highest probability of being the largest value, while gN has the lowest probability. This transformation causes too much distortion, and results in slow convergence as seen in Fig. 6 in sect. 4. To solve this, we chose another mapping in a similar way described P function xi 2 before, which is inspired by the fact of ( √P ) ≡ 1, as shown in (9), 2 xi

 xi , i = 1, 2, . . . , N g =r· s N  P 2  i xj , f= q P Nj=1 ρ   j=1 xj r= (Pmax − Pmin ) + Pmin N

(9)

where ρ is a factor used to control the distortion of transformation. Here ρ = 8.0 is chosen to get the smaller distortion as seen in Fig. 5. In our experiments, this results in a much faster convergence as seen in Fig. 6 in sect. 4.

4

Optimisation Results

Simulations were performed to test the efficacy of the approach and the potential of the method. The following is the list of simulation specifications.

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1

0.8

0.6

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0 0

0.5

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1.5

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Fig. 5. The marginal PDF of gi using (9), where N = 12, Pmax = 5.0, and Pmin = 0.01

Objective Function The objective function, or fitness function, is shown as (10). The main objective for our optimisation is to maximise system uplink capacity. We also need to minimise the RF power of base station to reduce interference, but this is not as important as the main objective, so we give it less weight, which is found by experiments. F itness = Capacity − 0.2 · (RF P ower f or all BSs).

(10)

Penalty function We compare our transformation-based method with a widely used constraints handling method, using penalty function.The penalty function is calculated by (11) and subtracted from the objective function. X P enalty = [P enalty value f or j − th pattern, Pj ], (11) 

P P exp( gi2 − Pmax ) , P gi2 > Pmax . 0, gi2 ≤ Pmax Simulation Configuration They are listed as follows: – 3000 traffic units and 100 base stations. Each base station has the capacity to serve 36 traffic units, i.e. T otal Capacity = 120% · T otal Demand; – 20% of traffic is uniformly distributed in the whole area; – Other 80% of traffic is distributed in 40 hot spots with normal distribution (the mean value for each hot spots, µ, is uniformly distributed over the whole area, and the standard deviation, σ = 0.2R). Results The optimisation was performed with three different methods. Some optimisation parameters are listed in Table 1. where Pj =

The optimisation results for two traffic scenarios are shown in Fig. 6 and Fig. 7. The horizontal lines in Fig. 6 indicate the system capacity of a conventional network, i.e. circular shapes, for those traffic scenarios. The results show that using the first mapping method (8) is much worse than using the second one

Constrained Coverage Optimisation for Mobile Cellular Networks

Table 1. Optimisation Parameters Penalty method First mapping Second mapping Population Size Generation Elite Rate Mapping Function BLX-0.5 Weight Distortion Weight RCGA Creep Weight Operator BLX-0.5 Rate Distortion Rate Creep Rate

1500 1000 0.1 N/A 2.0 0.5 0.05 0.2 0.3 0.005

1500 1000 0.1 (8) 2.0 0.5 0.05 0.2 0.3 0.005

1500 1000 0.1 (9) 2.0 0.5 0.05 0.2 0.3 0.005

2800

Scenario 1 Scenario 2

2700

System Uplink Capacity

2600 2500

Second mapping (9) 2400 2300

First mapping (8)

2200

Fixed (circular) pattern

2100 2000

Penality method

1900 1800 1700

0

250

500 Generation

750

1000

Fig. 6. System uplink capacity given by different optimisation methods

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Fig. 7. Part of the cellular coverage optimisation results

(9). It is sometimes even worse than not optimising. This highlights the importance of choosing a proper mapping function in transformation-based constraints handling methods. The results of using the penalty method are very bad. We believe this to be due to the small size of the feasible space in a huge search space. Most chromosomes at the first 300 generations are infeasible. Initially the higher capacity is obtained by covering more area than antennas can physically do, and so are not feasible, and the capacities here are not realisable capacities. After 300 generations, there are more and more feasible chromosomes, and the capacity start to increase, like the other two GA processes.

5

Other Applications

As a constraint handling technique for evolutionary computation, the transformationbased method has several interesting characteristics: – The transformation function is only relative to the constraints to be handled, and it is constructed before optimisation; – Other parts of the optimisation need not be aware of the transformation; – Proper distortion in the transformation can help finding better solutions. This method works very well when the feasible space is tiny, while the search space is huge, especially for high dimensional problems and equation constraints. Our application described in this paper is one example. Other two test cases from the paper [14] have also been evaluated. The first case is to maximise n √
n Y − → n · xi , G3( x ) = i=1

where

Pn

i=1

x2i = 1, and 0 ≤ xi ≤ 1 for 1 ≤ i ≤ n.

(12)

Constrained Coverage Optimisation for Mobile Cellular Networks

11

→ The function G3 has a global optimum solution at − x = ( √1n , ..., √1n ) and the value of the function in this point is 1. This case has been tested with an algorithm that redefined the crossover and mutation operators with the specific initialisation in their paper. To apply our transformation-based method, one possible mapping function for this case is shown as (13). ( xi = r · √P sni 2 , i = 1, 2, ..., n i=1 si , (13) f= r = 1.0 → → where − s ∈ S and − x ∈ F. The optimisation results showed similar performance to those in their paper. For the case n = 20, the system reached the value of 0.99 in less than 5000 generations (with population size of 30, probability of uniform crossover pc = 1.0, and probability of mutation pm = 0.06). However, our method is better at several aspects: 1) The specific initialisation is not necessary; 2) Standard genetic operators, such as uniform crossover and mutation, can be used without any modifications; 3) Usually, it requires more effort to redefine genetic operators than to find proper mapping functions. Another case is to minimise 2 → G11(− x ) = x21 + (x2 − 1) ,

(14)

where x2 − x21 = 0, and −1 ≤ xi ≤ 1, i = 1, 2. This case has been tested using hybrid methods in their paper, whilst in our test, it became quite simple. One possible mapping function, shown as (15), compresses the two-dimensional space into one-dimensional space.  x1 = (2s1 − 1) f= (15) 2 , x2 = (2s1 − 1) → → where − s ∈ S and − x ∈ F. The results present better performance than those in their paper. The system quickly reached the 0.75000455 in less than 30 generations in all runs. However, when the feasible space is rather large in the search space, or it is too difficult to find a proper mapping function, the transformation-based method is not worthwhile.

6

Conclusion

This paper has proposed and investigated the optimisation of schemes for smart antenna based dynamic cell size and shape control. Real-coded genetic algorithms are used to find the optimum cell size and shapes in the context of the whole cellular network and determine the potential of dynamic geographic load balancing scheme. This has proved important as it has allowed us to construct a benchmark for other real-time and distributed methods.

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We also describe an efficient methods to handle the power constraints in GA optimisation. The results show that a proper transformation is very important, as distortion in the mapping can slow down, rather than speed up convergence. The results from using the penalty method reflect the difficulty of using penalties when the solution space is very small relative to the unconstrained search space. The transformation-based methods can be applied into some of the constrained optimisations with a set of relative simple constraints.

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